Abstract : This thesis is devoted to the study ofthe stability of small stationary solutions of a nonlinear timedependent equation coming from relativistic quantum mechanics: thenonlinear Dirac equation.

In this study, non linear equations are viewed as small nonlinearperturbations of linear systems. A part of this thesis is hencedevoted to the study of linear problems. We prove that for a Diracoperator, with no resonance at thresholds nor eigenvalue atthresholds, the propagator satisfies propagation and dispersiveestimates. We also deduce smoothness estimates in the sense of Katoand Strichartz estimates.

With some ad hoc assumptions on the discrete spectrum of aDirac operator, we build small manifolds of stationary states. Thenwith small variations on these assumptions, we can highlight somestabilization process and orbital instability phenomena for somestationary states.